Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

52 Groundstate of a BEC in an optical lattice
5.2 Compressibility and effective coupling constant
From the solutions µ(s, gn) for the chemical potential, the compressibility κ of the system can
be immediately calculated using the relation
κ −1 = n ∂µ
(5.11)
∂n
It is useful to evaluate this quantity in order to understand better the combined effects of
lattice and interactions. In particular, analyzing the behavior of the compressibility helps to
study in more detail the dependence of the chemical potential on density in a lattice of fixed
depth. Knowing the form of this density-dependence is crucial to determine for example the
frequencies of collective oscillations, the density profile and the chemical potential obtained
when adding a harmonic trap to the optical lattice (see sections 5.4 and 9). In the uniform
system, the chemical potential is simply linear in the density µ = gn leading to an increase of
the inverse compressibility that is proportional to the density κ−1 = gn. In the presence of the
lattice, we expect deviations from this behaviour due to the localization of the particles near
the bottom of the well centers.
In Fig.5.4, the inverse compressibility κ −1 is plotted as a function of gn for different potential
depths s. As a general rule, the condensate becomes more rigid as gn/ER or s is increased.
Yet, in contrast to the uniform case, the monotonic growth of κ −1 is linear only at small
densities. At larger values of gn/ER, the slope of the curve tends to decrease and develop a
non-linear functional dependence.
At small values of gn where the inverse compressibility is approximately linear in gn, letus
denote the proportionality constant by ˜g such that
κ −1 =˜g(s)n. (5.12)
corresponding to the chemical potential
µ = µgn=0 +˜g(s)n, (5.13)
where µgn=0 depends on the lattice depth, but not on density. Since the condensate is more
rigid in the lattice relative to the uniform case, we have ˜g >g. Actually, ˜g is a monotonically
increasing function of s (see increase of the slopes at gn =0for increasing s in Fig. 5.4).
The quantity ˜g can be considered as an effective coupling constant: In a situation in which
Eq.(5.12) is valid, the compressibility of the condensate in the lattice with coupling constant
g is the same as the compressibility of a uniform condensate with coupling constant ˜g. So,as
far as the compressibility is concerned we can deal with the problem as if there was no lattice
by simply replacing g → ˜g. Below, we will see that this is a useful approach when describing
macroscopic properties, both static (see section 5.4) and dynamic (see section 9), which do
not require a detailed knowledge of the behavior on length scales of the order of the lattice
spacing d. In fact, the properties of the compressibility discussed in this section will prove
useful in devising a hydrodynamic formalism for condensates in a lattice.
The concept of an effective coupling constant ˜g is applicable when the influence of 2-body
interaction on the wavefunction ϕ(z) is negligible in the calculation of the chemical potential
(5.7). Provided this condition is fulfilled, we obtain using Eq.(5.7)
κ −1 = n ∂µ
d/2
= gnd |ϕgn=0(z)|
∂n 4 dz , (5.14)
−d/2